\(\int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx\) [2212]

Optimal result

Integrand size = 24, antiderivative size = 304 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=-\frac {3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{128 b^3 e^3}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^3 e^2}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {3 (b d-a e)^4 (2 A b e-B (b d+a e)) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{7/2} e^{7/2}} \]

[Out]

1/8*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(5/2)*(e*x+d)^(3/2)/b^2/e+1/5*B*(b*x+a)^(5/2)*(e*x+d)^(5/2)/b/e+3/128*(-a*e+
b*d)^4*(2*A*b*e-B*(a*e+b*d))*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(7/2)/e^(7/2)+1/64*(-a*e+b
*d)^2*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(3/2)*(e*x+d)^(1/2)/b^3/e^2+1/16*(-a*e+b*d)*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^
(5/2)*(e*x+d)^(1/2)/b^3/e-3/128*(-a*e+b*d)^3*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^3/e^3

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {81, 52, 65, 223, 212} \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\frac {3 (b d-a e)^4 (2 A b e-B (a e+b d)) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{7/2} e^{7/2}}-\frac {3 \sqrt {a+b x} \sqrt {d+e x} (b d-a e)^3 (2 A b e-B (a e+b d))}{128 b^3 e^3}+\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e)^2 (2 A b e-B (a e+b d))}{64 b^3 e^2}+\frac {(a+b x)^{5/2} \sqrt {d+e x} (b d-a e) (2 A b e-B (a e+b d))}{16 b^3 e}+\frac {(a+b x)^{5/2} (d+e x)^{3/2} (2 A b e-B (a e+b d))}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e} \]

[In]

Int[(a + b*x)^(3/2)*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

(-3*(b*d - a*e)^3*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(128*b^3*e^3) + ((b*d - a*e)^2*(2*A*b
*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x])/(64*b^3*e^2) + ((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*(a +
b*x)^(5/2)*Sqrt[d + e*x])/(16*b^3*e) + ((2*A*b*e - B*(b*d + a*e))*(a + b*x)^(5/2)*(d + e*x)^(3/2))/(8*b^2*e) +
(B*(a + b*x)^(5/2)*(d + e*x)^(5/2))/(5*b*e) + (3*(b*d - a*e)^4*(2*A*b*e - B*(b*d + a*e))*ArcTanh[(Sqrt[e]*Sqr
t[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(128*b^(7/2)*e^(7/2))

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {\left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right ) \int (a+b x)^{3/2} (d+e x)^{3/2} \, dx}{5 b e} \\ & = \frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {\left (3 (b d-a e) \left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right )\right ) \int (a+b x)^{3/2} \sqrt {d+e x} \, dx}{40 b^2 e} \\ & = \frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {\left ((b d-a e)^2 \left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{80 b^3 e} \\ & = \frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^3 e^2}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}-\frac {\left (3 (b d-a e)^3 \left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{320 b^3 e^2} \\ & = -\frac {3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{128 b^3 e^3}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^3 e^2}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {\left (3 (b d-a e)^4 \left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{640 b^3 e^3} \\ & = -\frac {3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{128 b^3 e^3}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^3 e^2}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {\left (3 (b d-a e)^4 \left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{320 b^4 e^3} \\ & = -\frac {3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{128 b^3 e^3}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^3 e^2}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {\left (3 (b d-a e)^4 \left (5 A b e-B \left (\frac {5 b d}{2}+\frac {5 a e}{2}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{320 b^4 e^3} \\ & = -\frac {3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{128 b^3 e^3}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^3 e^2}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {3 (b d-a e)^4 (2 A b e-B (b d+a e)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{7/2} e^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.05 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} \left (15 a^4 B e^4-10 a^3 b e^3 (4 B d+3 A e+B e x)+2 a^2 b^2 e^2 \left (5 A e (11 d+2 e x)+B \left (9 d^2+13 d e x+4 e^2 x^2\right )\right )+2 a b^3 e \left (5 A e \left (11 d^2+44 d e x+24 e^2 x^2\right )+B \left (-20 d^3+13 d^2 e x+136 d e^2 x^2+88 e^3 x^3\right )\right )+b^4 \left (10 A e \left (-3 d^3+2 d^2 e x+24 d e^2 x^2+16 e^3 x^3\right )+B \left (15 d^4-10 d^3 e x+8 d^2 e^2 x^2+176 d e^3 x^3+128 e^4 x^4\right )\right )\right )}{640 b^3 e^3}-\frac {3 (b d-a e)^4 (b B d-2 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{128 b^{7/2} e^{7/2}} \]

[In]

Integrate[(a + b*x)^(3/2)*(A + B*x)*(d + e*x)^(3/2),x]

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(15*a^4*B*e^4 - 10*a^3*b*e^3*(4*B*d + 3*A*e + B*e*x) + 2*a^2*b^2*e^2*(5*A*e*(11*d
+ 2*e*x) + B*(9*d^2 + 13*d*e*x + 4*e^2*x^2)) + 2*a*b^3*e*(5*A*e*(11*d^2 + 44*d*e*x + 24*e^2*x^2) + B*(-20*d^3
+ 13*d^2*e*x + 136*d*e^2*x^2 + 88*e^3*x^3)) + b^4*(10*A*e*(-3*d^3 + 2*d^2*e*x + 24*d*e^2*x^2 + 16*e^3*x^3) +
B*(15*d^4 - 10*d^3*e*x + 8*d^2*e^2*x^2 + 176*d*e^3*x^3 + 128*e^4*x^4))))/(640*b^3*e^3) - (3*(b*d - a*e)^4*(b*B
*d - 2*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(128*b^(7/2)*e^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1371\) vs. \(2(260)=520\).

Time = 1.08 (sec) , antiderivative size = 1372, normalized size of antiderivative = 4.51

method	result	size
default	\(\text {Expression too large to display}\)	\(1372\)

[In]

int((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/1280*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(544*B*a*b^3*d*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+880*A*((b*x+a)*(
e*x+d))^(1/2)*(b*e)^(1/2)*a*b^3*d*e^3*x+52*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2*b^2*d*e^3*x+52*B*((b*x+a)
*(e*x+d))^(1/2)*(b*e)^(1/2)*a*b^3*d^2*e^2*x+180*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*
d)/(b*e)^(1/2))*a^2*b^3*d^2*e^3+45*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2
))*a^4*b*d*e^4-30*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^2*d^2*e^
3-30*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b^3*d^3*e^2-120*A*ln(1/
2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*b^2*d*e^4-60*A*((b*x+a)*(e*x+d))^(1
/2)*(b*e)^(1/2)*b^4*d^3*e+45*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b
^4*d^4*e-120*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^4*d^3*e^2-60*A*
((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^3*b*e^4+320*A*b^4*e^4*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+30*A*ln(1/
2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^4*b*e^5+30*A*ln(1/2*(2*b*e*x+2*((b*x+
a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^5*d^4*e+30*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^4*e^4
+30*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*b^4*d^4-15*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a
*e+b*d)/(b*e)^(1/2))*b^5*d^5+352*B*a*b^3*e^4*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+352*B*b^4*d*e^3*x^3*((b*x
+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+480*A*a*b^3*e^4*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+480*A*b^4*d*e^3*x^2*((b
*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+16*B*a^2*b^2*e^4*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+16*B*b^4*d^2*e^2*x^2
*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+256*B*b^4*e^4*x^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-80*B*((b*x+a)*(e*x+
d))^(1/2)*(b*e)^(1/2)*a*b^3*d^3*e-20*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^3*b*e^4*x-20*B*((b*x+a)*(e*x+d))^
(1/2)*(b*e)^(1/2)*b^4*d^3*e*x+220*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2*b^2*d*e^3+220*A*((b*x+a)*(e*x+d))^
(1/2)*(b*e)^(1/2)*a*b^3*d^2*e^2-80*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^3*b*d*e^3+36*B*((b*x+a)*(e*x+d))^(1
/2)*(b*e)^(1/2)*a^2*b^2*d^2*e^2+40*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2*b^2*e^4*x+40*A*((b*x+a)*(e*x+d))^
(1/2)*(b*e)^(1/2)*b^4*d^2*e^2*x-15*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2
))*a^5*e^5)/b^3/e^3/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 1036, normalized size of antiderivative = 3.41 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\left [\frac {15 \, {\left (B b^{5} d^{5} - {\left (3 \, B a b^{4} + 2 \, A b^{5}\right )} d^{4} e + 2 \, {\left (B a^{2} b^{3} + 4 \, A a b^{4}\right )} d^{3} e^{2} + 2 \, {\left (B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} d^{2} e^{3} - {\left (3 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} d e^{4} + {\left (B a^{5} - 2 \, A a^{4} b\right )} e^{5}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (128 \, B b^{5} e^{5} x^{4} + 15 \, B b^{5} d^{4} e - 10 \, {\left (4 \, B a b^{4} + 3 \, A b^{5}\right )} d^{3} e^{2} + 2 \, {\left (9 \, B a^{2} b^{3} + 55 \, A a b^{4}\right )} d^{2} e^{3} - 10 \, {\left (4 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} d e^{4} + 15 \, {\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )} e^{5} + 16 \, {\left (11 \, B b^{5} d e^{4} + {\left (11 \, B a b^{4} + 10 \, A b^{5}\right )} e^{5}\right )} x^{3} + 8 \, {\left (B b^{5} d^{2} e^{3} + 2 \, {\left (17 \, B a b^{4} + 15 \, A b^{5}\right )} d e^{4} + {\left (B a^{2} b^{3} + 30 \, A a b^{4}\right )} e^{5}\right )} x^{2} - 2 \, {\left (5 \, B b^{5} d^{3} e^{2} - {\left (13 \, B a b^{4} + 10 \, A b^{5}\right )} d^{2} e^{3} - {\left (13 \, B a^{2} b^{3} + 220 \, A a b^{4}\right )} d e^{4} + 5 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} e^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{2560 \, b^{4} e^{4}}, \frac {15 \, {\left (B b^{5} d^{5} - {\left (3 \, B a b^{4} + 2 \, A b^{5}\right )} d^{4} e + 2 \, {\left (B a^{2} b^{3} + 4 \, A a b^{4}\right )} d^{3} e^{2} + 2 \, {\left (B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} d^{2} e^{3} - {\left (3 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} d e^{4} + {\left (B a^{5} - 2 \, A a^{4} b\right )} e^{5}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (128 \, B b^{5} e^{5} x^{4} + 15 \, B b^{5} d^{4} e - 10 \, {\left (4 \, B a b^{4} + 3 \, A b^{5}\right )} d^{3} e^{2} + 2 \, {\left (9 \, B a^{2} b^{3} + 55 \, A a b^{4}\right )} d^{2} e^{3} - 10 \, {\left (4 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} d e^{4} + 15 \, {\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )} e^{5} + 16 \, {\left (11 \, B b^{5} d e^{4} + {\left (11 \, B a b^{4} + 10 \, A b^{5}\right )} e^{5}\right )} x^{3} + 8 \, {\left (B b^{5} d^{2} e^{3} + 2 \, {\left (17 \, B a b^{4} + 15 \, A b^{5}\right )} d e^{4} + {\left (B a^{2} b^{3} + 30 \, A a b^{4}\right )} e^{5}\right )} x^{2} - 2 \, {\left (5 \, B b^{5} d^{3} e^{2} - {\left (13 \, B a b^{4} + 10 \, A b^{5}\right )} d^{2} e^{3} - {\left (13 \, B a^{2} b^{3} + 220 \, A a b^{4}\right )} d e^{4} + 5 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} e^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{1280 \, b^{4} e^{4}}\right ] \]

[In]

integrate((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

[1/2560*(15*(B*b^5*d^5 - (3*B*a*b^4 + 2*A*b^5)*d^4*e + 2*(B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 + 2*(B*a^3*b^2 - 6*A*
a^2*b^3)*d^2*e^3 - (3*B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (B*a^5 - 2*A*a^4*b)*e^5)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^
2*d^2 + 6*a*b*d*e + a^2*e^2 - 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b
*e^2)*x) + 4*(128*B*b^5*e^5*x^4 + 15*B*b^5*d^4*e - 10*(4*B*a*b^4 + 3*A*b^5)*d^3*e^2 + 2*(9*B*a^2*b^3 + 55*A*a*
b^4)*d^2*e^3 - 10*(4*B*a^3*b^2 - 11*A*a^2*b^3)*d*e^4 + 15*(B*a^4*b - 2*A*a^3*b^2)*e^5 + 16*(11*B*b^5*d*e^4 + (
11*B*a*b^4 + 10*A*b^5)*e^5)*x^3 + 8*(B*b^5*d^2*e^3 + 2*(17*B*a*b^4 + 15*A*b^5)*d*e^4 + (B*a^2*b^3 + 30*A*a*b^4
)*e^5)*x^2 - 2*(5*B*b^5*d^3*e^2 - (13*B*a*b^4 + 10*A*b^5)*d^2*e^3 - (13*B*a^2*b^3 + 220*A*a*b^4)*d*e^4 + 5*(B*
a^3*b^2 - 2*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^4*e^4), 1/1280*(15*(B*b^5*d^5 - (3*B*a*b^4 + 2*
A*b^5)*d^4*e + 2*(B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 + 2*(B*a^3*b^2 - 6*A*a^2*b^3)*d^2*e^3 - (3*B*a^4*b - 8*A*a^3*
b^2)*d*e^4 + (B*a^5 - 2*A*a^4*b)*e^5)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqr
t(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) + 2*(128*B*b^5*e^5*x^4 + 15*B*b^5*d^4*e - 10*(4*B*
a*b^4 + 3*A*b^5)*d^3*e^2 + 2*(9*B*a^2*b^3 + 55*A*a*b^4)*d^2*e^3 - 10*(4*B*a^3*b^2 - 11*A*a^2*b^3)*d*e^4 + 15*(
B*a^4*b - 2*A*a^3*b^2)*e^5 + 16*(11*B*b^5*d*e^4 + (11*B*a*b^4 + 10*A*b^5)*e^5)*x^3 + 8*(B*b^5*d^2*e^3 + 2*(17*
B*a*b^4 + 15*A*b^5)*d*e^4 + (B*a^2*b^3 + 30*A*a*b^4)*e^5)*x^2 - 2*(5*B*b^5*d^3*e^2 - (13*B*a*b^4 + 10*A*b^5)*d
^2*e^3 - (13*B*a^2*b^3 + 220*A*a*b^4)*d*e^4 + 5*(B*a^3*b^2 - 2*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a)*sqrt(e*x + d))
/(b^4*e^4)]

Sympy [F]

\[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\int \left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((b*x+a)**(3/2)*(B*x+A)*(e*x+d)**(3/2),x)

[Out]

Integral((A + B*x)*(a + b*x)**(3/2)*(d + e*x)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2589 vs. \(2 (260) = 520\).

Time = 0.67 (sec) , antiderivative size = 2589, normalized size of antiderivative = 8.52 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^(3/2)*(B*x+A)*(e*x+d)^(3/2),x, algorithm="giac")

[Out]

1/1920*(80*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*
a*b^5*e^4)/(b^7*e^4)) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)/(b^7*e^4)) - 3*(b^3*d^3 + a*b^2*d^2*e
+ 3*a^2*b*d*e^2 - 5*a^3*e^3)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b
*e)*b*e^2))*A*d*abs(b) + 10*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 +
(b^12*d*e^5 - 25*a*b^11*e^6)/(b^14*e^6)) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)/(b^14*e^6)) +
3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)/(b^14*e^6))*sqrt(b*x + a) + 3*(5*
b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*e^4)*log(abs(-sqrt(b*e)*sqrt(b*x + a) +
sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b^2*e^3))*B*d*abs(b) - 1920*((b^2*d - a*b*e)*log(abs(-sqrt(b*
e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b*e) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(
b*x + a))*A*a^2*d*abs(b)/b^2 + 160*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a
)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)/(b^7*e^4)) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)/(b^7*e^4)) -
3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a
)*b*e - a*b*e)))/(sqrt(b*e)*b*e^2))*B*a*d*abs(b)/b + 10*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(
b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)/(b^14*e^6)) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163
*a^2*b^11*e^6)/(b^14*e^6)) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)/(b^14
*e^6))*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*e^4)*log(abs
(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b^2*e^3))*A*e*abs(b) + (sqrt(b^2*
d + (b*x + a)*b*e - a*b*e)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*d*e^7 - 41*a*b^19*e^8)/(b^23*
e^8)) - (7*b^21*d^2*e^6 + 26*a*b^20*d*e^7 - 513*a^2*b^19*e^8)/(b^23*e^8)) + 5*(7*b^22*d^3*e^5 + 19*a*b^21*d^2*
e^6 + 37*a^2*b^20*d*e^7 - 447*a^3*b^19*e^8)/(b^23*e^8))*(b*x + a) - 15*(7*b^23*d^4*e^4 + 12*a*b^22*d^3*e^5 + 1
8*a^2*b^21*d^2*e^6 + 28*a^3*b^20*d*e^7 - 193*a^4*b^19*e^8)/(b^23*e^8))*sqrt(b*x + a) - 15*(7*b^5*d^5 + 5*a*b^4
*d^4*e + 6*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 35*a^4*b*d*e^4 - 63*a^5*e^5)*log(abs(-sqrt(b*e)*sqrt(b*x + a
) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b^3*e^4))*B*e*abs(b) + 80*(sqrt(b^2*d + (b*x + a)*b*e - a
*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)/(b^7*e^4)) - 3*(b^7*d^2*e^2 + 2
*a*b^6*d*e^3 - 11*a^2*b^5*e^4)/(b^7*e^4)) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*log(abs(-sqr
t(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b*e^2))*B*a^2*e*abs(b)/b^2 + 160*(sqrt
(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)/(b^7*
e^4)) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)/(b^7*e^4)) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2
- 5*a^3*e^3)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b*e^2))*A*a*
e*abs(b)/b + 20*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5
- 25*a*b^11*e^6)/(b^14*e^6)) - (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)/(b^14*e^6)) + 3*(5*b^14*d
^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)/(b^14*e^6))*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*
a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 - 35*a^4*e^4)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d +
(b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b^2*e^3))*B*a*e*abs(b)/b + 480*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*
x + 2*a + (b*d*e - 5*a*e^2)/e^2)*sqrt(b*x + a) + (b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*log(abs(-sqrt(b*e)*sqrt
(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*e))*B*a^2*d*abs(b)/b^3 + 960*(sqrt(b^2*d + (b*x +
a)*b*e - a*b*e)*(2*b*x + 2*a + (b*d*e - 5*a*e^2)/e^2)*sqrt(b*x + a) + (b^3*d^2 + 2*a*b^2*d*e - 3*a^2*b*e^2)*l
og(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*e))*A*a*d*abs(b)/b^2 + 480*
(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + 2*a + (b*d*e - 5*a*e^2)/e^2)*sqrt(b*x + a) + (b^3*d^2 + 2*a*b^2*
d*e - 3*a^2*b*e^2)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*e))*A*a
^2*e*abs(b)/b^3)/b

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\int \left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2} \,d x \]

[In]

int((A + B*x)*(a + b*x)^(3/2)*(d + e*x)^(3/2),x)

[Out]

int((A + B*x)*(a + b*x)^(3/2)*(d + e*x)^(3/2), x)